A085831 a(n) = Sum_{k=1..2^n} d(k) where d(n) = number of divisors of n (A000005).

1, 3, 8, 20, 50, 119, 280, 645, 1466, 3280, 7262, 15937, 34720, 75108, 161552, 345785, 736974, 1564762, 3311206, 6985780, 14698342, 30850276, 64607782, 135030018, 281689074, 586636098, 1219788256, 2532608855, 5251282902, 10874696106, 22493653324, 46475828418

Offset: 0

Author

Robert G. Wilson v, Sep 21 2000

Keywords

• nonn

Links

• Chai Wah Wu, Table of n, a(n) for n = 0..82 (terms 0..64 from Donovan Johnson)

Crossrefs

Cf. A006218, A050226, A085567, A085829, A057494.

Programs

Mathematica

k = s = 0; Do[ While[ k < 2^n, k++; s = s + DivisorSigma[ 0, k ]]; Print[s], {n, 0, 29} ]

PARI

a(n) = sum(k=1, 2^n, numdiv(k)); \\ Michel Marcus, Oct 10 2021

Python

from math import isqrt
def A085831(n): return (lambda m, r: 2*sum(r//k for k in range(1, m+1))-m*m)(isqrt(2**n),2**n) # Chai Wah Wu, Oct 08 2021